At \(\theta =1.05^\circ \) , the first order gradient of fermion fields or of the atomic displacement fields in \(H_{intra}\)  are of the same order as the contact terms in \(H_{inter}\)  [1]}. The second order in gradients intra-layer terms are, in turn, of the same order as the first order in gradients inter-layer terms. This pattern continues for the higher order terms. As shown in the Fig. REF , the disagreement between the first order continuum Hamiltonian spectrum and the exact tight binding spectrum is \(\sim 10\) meV, i.e. of the order of the narrow bandwidth. On the other hand, including the second order terms in the continuum Hamiltonian improves the agreement significantly as seen in the Figs. REF , with a nearly perfect agreement throughout the moire Brillouin zone; the largest disagreement is near the \(\Gamma \)  point where there is at most \(0.7\) meV difference for the model of Ref. [2]} and at most \(0.3\) meV difference for the model of Ref. [3]}.
